Algebra 1 linear functions slope

Explanations of Slope from 9th Graders

Today’s submitter asked the following question:


 Here are a selection of student responses:

  • Slope is how far apart the two points are. You can use the idea of rise over run. You rise a certain amount and then run right or left depending ona negative or positive.  
  • Slope is how you measure two or more points or equations on a graph. 
  • It’s the y-intercept divided by the x intercept and intercept is the coordinates. 
  • Slope is the change in the dependent (which can be anything) over the independent (which can be anything).
  • Slope is one point of a line on a graph minus another point on the same line in that graph. 
  • The slope is the difference of the dependent variable (y) over the independent variable (x).
  • Slope is the rate of change between the dependent and the independent. If the dependent went up by 2 every time x, the independent, went up by 1, then the slope would be y over x which in this case is 2. 
  • Slope is when you take one point on a graph and a second one and see the difference between them and the numbers whatever y go over the numbers of the difference of x and that’s your answer. 
  • Slope is the rate of change in an equation, graph or story. For example, 2x -4 = y, take two possible solutions the rate of change is 2. 
  • Slope is the unit that you “go up by”. Let’s say you buy 4 apples for $4. The slope is 1 because it always goes up by one, 6 apples would be $5. 
  • Slope is the rate of change in an equation. 
  • The rate of change, between an independent and depending things.
  • You have two points so all you do is see how far apart they are. We see that it goes from (4, 2) to (2,1) so it’s 2/1.
  • Slope shows how a number increases or decreases and by how much. 
  • A slope is the change of rate in a problem for example, a man gets paid $5 an hour How much does he get after working for 3 hours? He gets $15, so 5 is the slope. 
  • Slope is when you have 2 numbers and find out what the answer will be. 

There’s a lot to work on here. In the comments, pick a student and dig into their understanding of slope. And also, can you infer what the approach of the unit was from the students’ responses? How could the unit be improved?

Algebra 1 Quadratics

Quadratics Quiz

As we start a new week of Math Mistakes, if you have any pictures of student work that you find interesting in any way, please send it in. 


Today’s student has a couple of ideas worth drawing out. Do so in the comments.


Geometry Quadrilaterals

False Symmetry


How do you get kids to stop seeing symmetry when it just isn’t there?

Fractions Numbers & Operations -- Fractions

“Fractions are really small things.”

Today isn’t a sample of student work, exactly. It’s an interview with a kid.




[Source: How Students Learn: Mathematics in the Classroom, p.314]

3.NF.2 Fractions

How big is three-eighths?

In the student’s work below, what is “it” and why does the student think that matters?


[Source: ]

Algebra 2 F-BF.1 Functions

Composition of Functions

This kid clearly knows some of the basics of composing functions. But check out that last line.




What is going through this kid’s brain, and how would you respond?

Algebra 2 G-SRT.10 Law of Cosines Law of Sines Trigonometry

Trigonometry – Find the missing angle


My sense is that this mistake isn’t as interesting as the rest, but it’s a pretty common one that I see in Trigonometry. The question is, what sort of activity would help this student out?

A-SSE.2 Algebra 1 factoring

What does he think “factoring” means?


What does this kid think “factoring” means? And, at least to me, the more interesting question is: where did that conception of “factoring” come from? Tease it out in the comments.

8.EE.1 Algebra 1 exponents

Powers Puzzler


What is going through this student’s brain? What would you say to the kid to help out?

Let me know if you’re having trouble reading today’s submission. If image quality continues to be a problem then I’ll experiment with typing the student work up.

Algebra 2 Rational Expressions

Not-so-simple simplifying

What’s the student thinking, and how can we help?